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How can we reason about “if P then Q” or “P only if Q” statements in propositional logic?

Here's the table for If-then

 _____ ___________
| A B |  (A ⊃ B)  |
| 0 0 |    1      |
| 0 1 |    1      |
| 1 0 |    0      |
| 1 1 |    1      |
|_____|___________|

Let A = There's a God. and B = There's a human. So, If there's a God then there's a human.

Acording to truth table,

When A = 1 and B = 1 or A = 0 and B = 0, the result is understandable,

But, the other two statements for example:

"If there's not a God then there's a human" is true. Why is this? Also,
"If there's a God then there's not a human" is the only statement that is false.

How does it make sense?

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The various truth assignments don't modify the proposition "If there is God, then there's a human"; they're assignments to whether or not there are gods or humans, and the truth value of A ⊃ B represents whether or not the hypothetical world being described — with or without gods, and with or without humans — is consistent with the statement that if there's a god, then there's a human.

It might help you to rephrase this as "There is God only if there is a human", which is equivalent, and can be easily understood as a constraint on the conditions in which God can exist.

  • A=0, B=0 ⇒ there's no God and no human; as there's no God, the constraints on its existence is not violated, so A ⊃ B = 1.

  • A=0, B=1 ⇒ there's no God, but there are humans; similarly to the above, the constraint on God-existence is not violated, so A ⊃ B = 1.

  • A=1, B=0 ⇒ there's God without humans; this violates the constraint, so A ⊃ B = 0.

  • A=1, B=1 ⇒ there's God, but there are also humans; the necessary condition for God's existence is met, so A ⊃ B = 1.

In the first two cases, A ⊃ B is often described as vacuously true, as the premise of God existing is false in those cases — equivalently, the condition which is restricted by the consequences fails to hold anyway, so the constraint is satisfied by that very failure.

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  • Just a practical question: how do you put symbols in the answer? Copy/paste from another place?
    – Koeng
    Oct 17, 2012 at 14:56
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    @Koeng: I know the secret codes! ⇒ = "⇒", ⊃ = "⊃", — = "—". The only way to put the symbols in the comments, however, is copy and paste.
    – Niel de Beaudrap
    Oct 17, 2012 at 14:59
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    @Niel - Indeed helpful, and probably something we should put in our faq! ^_^
    – stoicfury
    Oct 17, 2012 at 16:38
  • @NieldeBeaudrap really love this answer -- since we're now getting a lot of questions around this topic, is there any chance I might be able to persuade you to consider formulating a nice and general question about this and 'canonical' answer?
    – Joseph Weissman ♦
    Oct 30, 2012 at 15:51
  • @JosephWeissman: Perhaps! What sorts of features would you be looking for in the model question, which aren't already present here? I imagine that the model answer would involve different glosses for A→B (A⊃B) in classical logic, with a colourful (but perhaps not theologically informed) example to illustrate it.
    – Niel de Beaudrap
    Oct 30, 2012 at 16:19
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Keep in mind that the truth table for implication only shows you what the combination of the different values for the statements lead as a conclusion for the implication. It's about the validity of the conclusions, not about the truth of the statements. You could use "horses like bananas => microwaves are rock stars" and it wouldn't matter. You can't read the truth table as if it were saying something about the "sense" of the statement.

So, that said, let's talk about the cases you have doubt:

  1. "If there's not a God then there's a human" - this is part of the cases where the truth table says that if A is false, than your implication doesn't deduce anything at all. Notice that both (A,B) = (0,1) and (0,0) are true. The value of A => B can also be seen as ~A V B, which includes: if A is false, the implication is true.

  2. "If there's a God then there's not a human" - this case is the only one false because it is the only one that doesn't allow A => B to be true. Again, it's a matter of validity, not a matter of the sense of the statements. If you define A as "there's a God" and define B as "there's a human", then when you ask for the truth values of A => B, it will be false if the case is A => ~B, i.e., "If there's a God then there's not a human".

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"If A Then B" implies nothing about the world if A is false. So the statement can't be disproved if A is false. Technically, I suppose it's more "undefined" than "True" when A is false - but truth tables don't usually allow for an undefined value.

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this question is a play with logic and selection & assignment of words to the variables. What would you argue if you reverse the same? A becomes human and B is God? Or what is A is "Evil", will that going to conclude If Evil, then Human?

Here "God" is not a proven entity or Evil. They are conceptual idealism. One can be taught Evil is good and God is bad. So it is up to the receiver how to perceive those entities.

In my humble opinion, at-least to use logical proposition, the selection of items should be a fact. If your selections are not facts then logical proposition cannot hold responsible for your conclusion.

true and false are mere facts and nothing else. so your selection should also match with them to derive a new axiom or fact.

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